Higher and derived stacks: a global overview

Abstract

These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.

51 pages. One reference added. More corrections (comments are still very much welcome)

References in corpus (4)

• Descente pour les n-champs (Descent for n-stacks)
• Schematic homotopy types and non-abelian Hodge theory
• A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi
• A Homotopy Theory for Stacks

Cited by in corpus (39)

• Twisted differential String and Fivebrane structures
• A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
• Legendrian knots and constructible sheaves
• Presenting higher stacks as simplicial schemes
• The Character Theory of a Complex Group
• Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds
• Quasi-Hamiltonian reduction via classical Chern-Simons theory
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Proper local complete intersection morphisms preserve perfect complexes
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• D-manifolds, d-orbifolds and derived differential geometry: a detailed summary
• A note on Chern character, loop spaces and derived algebraic geometry
• K-theory and G-theory of derived algebraic stacks
• Loop Spaces and Langlands Parameters
• Lectures on shifted Poisson geometry
• Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory
• Moduli of $p$-adic representations of a profinite group
• Spontaneous symmetry breaking: a view from derived geometry
• The homology of moduli stacks of complexes
• Enumerative invariants and wall-crossing formulae in abelian categories
• A period map for global derived stacks
• Non-archimedean quantum K-invariants
• The global derived period map
• Dieudonné Theory via Cohomology of Classifying Stacks
• Differential characters as stacks and prequantization
• An introduction to derived (algebraic) geometry
• Derived ${\mathcal{O}}_k$-adic geometry and derived Raynaud localization theorem
• Hall algebras and Donaldson-Thomas invariants
• Higher Galois for Segal Topos and Natural Phenomena
• A Simple Construction of Derived Representation Schemes
• Moduli stack of oriented formal groups and periodic complex bordism
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• Representability of cohomology of finite flat abelian group schemes
• A Whirlwind Tour of the World of $(\infty ,1)$-categories
• Motifs in Derived Algebraic Geometry
• A Physical Origin for Singular Support Conditions in Geometric Langlands Theory
• Higher theta series for unitary groups over function fields
• Segal-type models of higher categories
• Unifying derived deformation theories